↑

Properties

Label 12.4.5743027167914169.1
Degree $12$
Signature $[4, 4]$
Discriminant $3^{6}\cdot 7^{10}\cdot 167^{2}$
Root discriminant $20.57$
Ramified primes $3, 7, 167$
Class number $1$
Class group Trivial
Galois group $C_2\times C_2^2\wr C_2:C_3$ (as 12T87)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![43, -145, 126, -85, -23, 21, -34, 14, -4, 6, 0, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 3*x^11 + 6*x^9 - 4*x^8 + 14*x^7 - 34*x^6 + 21*x^5 - 23*x^4 - 85*x^3 + 126*x^2 - 145*x + 43)
 
gp: K = bnfinit(x^12 - 3*x^11 + 6*x^9 - 4*x^8 + 14*x^7 - 34*x^6 + 21*x^5 - 23*x^4 - 85*x^3 + 126*x^2 - 145*x + 43, 1)
 

Normalized defining polynomial

\( x^{12} - 3 x^{11} + 6 x^{9} - 4 x^{8} + 14 x^{7} - 34 x^{6} + 21 x^{5} - 23 x^{4} - 85 x^{3} + 126 x^{2} - 145 x + 43 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $12$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(5743027167914169=3^{6}\cdot 7^{10}\cdot 167^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.57$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $3, 7, 167$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{10} a^{9} - \frac{3}{10} a^{8} - \frac{1}{10} a^{7} - \frac{1}{5} a^{6} - \frac{3}{10} a^{4} - \frac{1}{5} a^{3} + \frac{2}{5} a^{2} + \frac{1}{5} a + \frac{3}{10}$, $\frac{1}{10} a^{10} - \frac{1}{2} a^{7} + \frac{2}{5} a^{6} - \frac{3}{10} a^{5} - \frac{1}{10} a^{4} - \frac{1}{5} a^{3} + \frac{2}{5} a^{2} - \frac{1}{10} a - \frac{1}{10}$, $\frac{1}{641627270} a^{11} - \frac{8883657}{320813635} a^{10} - \frac{7615498}{320813635} a^{9} + \frac{34610203}{641627270} a^{8} + \frac{26623843}{64162727} a^{7} + \frac{48052613}{641627270} a^{6} - \frac{176739019}{641627270} a^{5} + \frac{196452}{64162727} a^{4} - \frac{145891143}{320813635} a^{3} + \frac{224775809}{641627270} a^{2} - \frac{207611119}{641627270} a + \frac{134143108}{320813635}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1562.37870848 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^2\wr C_2:C_3$ (as 12T87):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 192
The 20 conjugacy class representatives for $C_2\times C_2^2\wr C_2:C_3$
Character table for $C_2\times C_2^2\wr C_2:C_3$

Intermediate fields

\(\Q(\sqrt{21}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{21})^+\)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 12 siblings: data not computed
Degree 16 siblings: data not computed
Degree 24 siblings: data not computed
Degree 32 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/5.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$3$3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$7$7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
$167$$\Q_{167}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{167}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{167}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{167}$$x + 2$$1$$1$$0$Trivial$[\ ]$
167.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
167.2.1.1$x^{2} - 167$$2$$1$$1$$C_2$$[\ ]_{2}$
167.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
167.2.1.1$x^{2} - 167$$2$$1$$1$$C_2$$[\ ]_{2}$